DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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Characters and normalized Fourier coefficients

Definition

Use the cube and dot product of The BLR linearity test over F_2. For $a, x \in F_{2}^{n}$ , $n \geq 0$ , the real-valued character is $χ_{a} (x) = (- 1)^{a \cdot x}$ . For $h : F_{2}^{n} \to R$ , define $\hat{h} (a) = E_{x} h (x) χ_{a} (x) = 2^{- n} \sum_{x \in F_{2}^{n}} h (x) χ_{a} (x) .$ The expectation symbol is just the uniform arithmetic mean. Products, sums of coefficients, and these averages are in $R$ ; only the dot product and vector addition are modulo two. In particular $χ_{0} = 1$ and $\hat{h} (0)$ is the mean of $h$ .

Depends on

The BLR linearity test over F_2

Used by

Character orthogonality, inversion and Parseval Lemma

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Arora and Barak, Computational Complexity, January 2007 web draft, §19.3.1, pp.388–389 (standard reference, not scraped)